for this to be an equality we need the left side to be a positive integer , because the right side we know is a positive integer , (k , j are both positive integers so the product of two positive integers is another positive integer)

so then having k+ 3/(a+1)=kj and rewriting it by subtracting k to the other side : 3/(a+1) = k(j-1) wouldn't that alright since k is a positive and j is a positive integer and subtracting 1 from a positive integer is still positive so the right side would be a positive integer

1. k(j-1) is integer since integers are closed under multiplication
2. k(j-1) is positive because it equals to 3/(a+1) , and we assumed a > 0
3. 3/(a+1) must be a positive integer because it equals k(j-1) which we have shown
earlier is a positive integer
4. this only leaves one option for a, for a solution to exist. a = 2,

so the proof goes, a>0 , b>0
b divides b+3. so (b+3) / b is a positive integer.
but (b+3)/b = 1 + 3/b This leaves only two options for b, either b = 1 , or b = 3.
But b=1 leads to a = 0 from the second condition that a+1 divides b. This contradicts a >0. If b = 3 , then a = 2 works, 2+1 | 3 , and everyone is happy